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Full text of "Treatise On Analysis Vol-Ii"

340       XV   NORMED ALGEBRAS AND SPECTRAL THEORY

(15.4.15) Let A be a commutative star algebra with unit element e^Q.
Suppose that there exists x0 e A such that the subalgebra generated by e, x0 and
x* is dense in A. Then the mapping x*-*x(*o) is a homeomorphism o/X(A)
onto SpA(x0)-

As in (15.3.6) we see that A is separable, and that it is enough to prove
that the relation XI(XQ) = X2(*o) for two characters &, %2 implies that Xi = Xf>.
Now, from (15.4.14) we deduce that Xi(*o) = tofro) and thence that
Xi(P(xo,x$) = X2(*(x<x$) for a11 polynomials PeC[X,Y]. Since by
hypothesis the P(*0 > **) are dense in A, and since %19 I2 are continuous on A,
the result follows.

PROBLEMS

1.    Let A be a normed algebra endowed with a continuous involution x\->x*. If
||#||! = sup(||*||, ||jc*||), show that with the norm \\x\\i the algebra A becomes a
normed algebra with involution, and that the norms ||x|| and \\x\\ i are equivalent.

2.    Let A be a commutative Banach algebra without radical (Section 15.2, Problem 7).
Show that every involution on A is continuous (use Problem 20 of Section 15.3).

3.    Let A be a commutative Banach algebra with involution, having a unit element. A
character % of A is said to be hermitian if xC**) = x(*) fr aU xe A. Show that every
character of A is hermitian if and only if, for each x e A, the element 1 + xx* is
invertible in A. (If every character of A is hermitian, then #(1 H- xx*) is a function
which is everywhere >0 on X(A); use (15.3.4). If a character x of A is not hermitian,
then there exists a self-adjoint y e A such that xOO = ' whence %(1 + yy*) ~ 0,
and 1 + yy* is not invertible.) Hence give an example of a commutative Banach
algebra with involution, having a unit element, for which there exist nonhermitian
characters.

4.    (a)   If A is a star algebra with unit element, then ||*||2 =p(x*x) for all x e A (prove
this first when x is self-adjoint). Deduce that A is without radical.

oo

(b)   Let A be the algebra of Problem 2 of Section 15.1. If x =  nTn, define x* to be

n = 0

oo    ,

2 snT". This defines an involution on A for which A is a commutative Banach algebra

n = 0

with involution, having a unit element, and such that the only character of A (Section
15.3, Problem 2) is hermitian.

5.    With respect to the involution/*(/)=/(/) on the Beurling algebra (Section 15.1,
Problem 4 and Section 15.2, Problem 6), show that all the characters are hermitian.

6.    On the algebra A of Problem 7 of Section 15.3, consider the involution induced by
/i-*/, and extend this involution to A by putting e* = e. Then A is a Banach algebra
with involution. Find its hermitian characters. Problem 7) is